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In a wide variety of imaging applications (especially medical imaging), we obtain a partial set or subset of the Fourier transform of an image. From these Fourier measurements, we want to reconstruct the entire original image. Convex optimization is a powerful, recent solution to this problem. Unfortunately, convex optimization in its myriad of implementations is computationally expensive and may be impractical for large images or for multiple images. Furthermore, some of these techniques assume that the image has a sparse gradient (i.e., that the gradient of the image consists of a few nonzero pixel values) or that the gradient is highly compressible. In this paper, we demonstrate that we can recover such images with GradientOMP, an efficient algorithm based upon Orthogonal Matching Pursuit (OMP), more effectively than with convex optimization. We compare both the qualitative and quantitative performance of this algorithm to the optimization techniques.